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Platonic numbers

(Redirected from Dodecahedral numbers)

The five regular convex polyhedra (3-dimensional regular convex solids, known as the 5 Platonic solids), are

The tetrahedron is self-dual, the cube and the octahedron are duals, and the dodecahedron and icosahedron are duals. (Dual pairs have same number of edges and have vertices corresponding to faces of each other.)

Number of vertices, edges and faces of the 5 Platonic solids:

• A063723 Number of vertices in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).
• A063722 Number of edges in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).
• A053016 Number of faces of Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).
The Platonic numbers (A053012) are the numbers of dots in a layered geometric arrangement into one of the 5 Platonic solids. The platonic numbers start with one initial dot (for
 n = 1
), then with one dot at each vertex of a given Platonic solid (for
 n = 2
), with each of the following layers growing out of the initial vertex with one more dot per edge than the preceding layer, and where overlapping dots (the dot at the initial vertex and the dots on all the edges sharing that initial vertex) are counted only once.

The 5 types of Platonic numbers (by increasing number of vertices) are:

• A000292 Tetrahedral (or triangular pyramidal) numbers:
( n + 2 3  ) =  n (n + 1) (n + 2) 6
.
• A005900 Octahedral numbers:
 2n 3 + n 3
=  n (2n 2 + 1) 3
.
• A000578 The cubes:  n 3
.
• A006564 Icosahedral numbers:
 n (5n 2  −  5n + 2) 2
.
• A006566 Dodecahedral numbers:
 n (9n 2  −  9n + 2) 2
=  n (3n  −  1) (3n  −  2) 2
=  3n (3n  −  1) (3n  −  2) 6
= (  3n 3  )
.

All figurate numbers are accessible via this structured menu: Classifications of figurate numbers.

Formulae

The
 n
th Platonic
 N2
-hedral number (having
 N0
vertices) is given by the formulae:
 P  (3) N0(n) = (  n + 2 3  ) + (N0  −  4) (  n + 1 3  ) + 0 (  n 3  )
, for the (self dual) tetrahedral (
 N0 = 4
) numbers;
 P  (3) N0(n) = (  n + 2 3  ) + (N0  −  4) (  n + 1 3  ) + 1 (  n 3  )
for the (dual pair) octahedral (
 N0 = 6
) and hexahedral (cubic) (
 N0 = 8
) numbers;
P  (3)N0(n) = ( n + 2 3  ) + (N0  −  4) ( n + 1 3  ) + (
 N0 2
) ( n 3  )
for the (dual pair) icosahedral (
 N0 = 12
) and dodecahedral (
 N0 = 20
) numbers.
where
 N0
is the number of 0-dimensional elements (vertices
 V
),
 N1
is the number of 1-dimensional elements (edges
 E
),
 N2
is the number of 2-dimensional elements (faces
 F
) of the polyhedron.
 N0 = (2 r  −  0 r ) + 4
is the number of vertices of the Platonic solid, where
 r ∈ {0, 1, 2, 3, 4}
is the rank of the Platonic solid (by increasing number of vertices).

Platonic roots

Tetrahedral roots

The tetrahedral roots of
 n
are defined as the roots
 r
of the cubic equation
${\begin{array}{l}\displaystyle {n={\frac {r\,(r+1)(r+2)}{6}}={\frac {r^{3}}{6}}+{\frac {r^{2}}{2}}+{\frac {r}{3}},}\end{array}}$ hence

${\begin{array}{l}\displaystyle {r^{3}+3r^{2}+2r-6n=0.}\end{array}}$ Cube roots

The cube roots of
 n
are defined as the roots
 r
of the cubic equation
${\begin{array}{l}\displaystyle {r^{3}-n=0,}\end{array}}$ yielding

${\begin{array}{l}\displaystyle {r={\sqrt[{3}]{n}}.}\end{array}}$ Octahedral roots

The octahedral roots of
 n
are defined as the roots
 r
of the cubic equation
${\begin{array}{l}\displaystyle {n={\frac {r\,(2r^{2}+1)}{3}},}\end{array}}$ hence

${\begin{array}{l}\displaystyle {2r^{3}+r-3n=0.}\end{array}}$ Dodecahedral roots

The dodecahedral roots of
 n
are defined as the roots
 r
of the cubic equation
${\begin{array}{l}\displaystyle {n={\frac {r\,(9r^{2}-9r+2)}{2}},}\end{array}}$ hence

${\begin{array}{l}\displaystyle {9r^{3}-9r^{2}+2r-2n=0.}\end{array}}$ Icosahedral roots

The icosahedral roots of
 n
are defined as the roots
 r
of the cubic equation
${\begin{array}{l}\displaystyle {n={\frac {r\,(5r^{2}-5r+2)}{2}},}\end{array}}$ hence

${\begin{array}{l}\displaystyle {5r^{3}-5r^{2}+2r-2n=0.}\end{array}}$ Descartes–Euler (convex) polyhedral formula

Descartes–Euler (convex) polyhedral formula:

${\begin{array}{l}\displaystyle {{\sum _{i=0}^{2}(-1)^{i}N_{i}}=N_{0}-N_{1}+N_{2}=V-E+F=2,}\end{array}}$ where
 N0
is the number of 0-dimensional elements (vertices
 V
),
 N1
is the number of 1-dimensional elements (edges
 E
),
 N2
is the number of 2-dimensional elements (faces
 F
) of the polyhedron.

Recurrence relation

${\begin{array}{l}\displaystyle {P_{N_{0}}^{(3)}(n)=4P_{N_{0}}^{(3)}(n-1)-6P_{N_{0}}^{(3)}(n-2)+4P_{N_{0}}^{(3)}(n-3)-P_{N_{0}}^{(3)}(n-4),\quad n>3,}\end{array}}$ with initial conditions

{\begin{array}{l}\displaystyle {\begin{aligned}P_{N_{0}}^{(3)}(0)&=0,\\P_{N_{0}}^{(3)}(1)&=1,\\P_{N_{0}}^{(3)}(2)&=N_{0},\\P_{N_{0}}^{(3)}(3)&=\;?.\end{aligned}}\end{array}} Ordinary generating function

${\begin{array}{l}\displaystyle {G_{\{P_{N_{0}}^{(3)}(n)\}}(x)={\frac {x\,(1+(N_{0}-4)\,x+0\,x^{2})}{(1-x)^{4}}}={\frac {x\,(1+(2^{r}-0^{r})\,x+0\,x^{2})}{(1-x)^{4}}},}\end{array}}$ for the (self dual) tetrahedral (rank
 r = 0, N0 = 4
vertices) numbers;
${\begin{array}{l}\displaystyle {G_{\{P_{N_{0}}^{(3)}(n)\}}(x)={\frac {x\,(1+(N_{0}-4)\,x+1\,x^{2})}{(1-x)^{4}}}={\frac {x\,(1+(2^{r}-0^{r})\,x+1\,x^{2})}{(1-x)^{4}}},}\end{array}}$ for the (dual pair) octahedral (rank
 r = 1, N0 = 6
vertices) and hexahedral (cubic) (rank
 r = 2, N0 = 8
vertices) numbers;
${\begin{array}{l}\displaystyle {G_{\{P_{N_{0}}^{(3)}(n)\}}(x)={\frac {x\,(1+(N_{0}-4)\,x+\left({\tfrac {N_{0}}{2}}\right)x^{2})}{(1-x)^{4}}}={\frac {x\,(1+(2^{r}-0^{r})\,x+\left({\tfrac {N_{0}}{2}}\right)x^{2})}{(1-x)^{4}}},}\end{array}}$ for the (dual pair) icosahedral (rank
 r = 3, N0 = 12
vertices) and dodecahedral (rank
 r = 4, N0 = 20
vertices) numbers.
 N0 = (2 r  −  0 r )  +  4
is the number of vertices of the Platonic solid, where
 r ∈ {0, 1, 2, 3, 4}
is the rank of the Platonic solid (by increasing number of vertices).

Exponential generating function

${\begin{array}{l}\displaystyle {E_{\{P_{N_{0}}^{(3)}(n)\}}(x)=\;?.}\end{array}}$ For the 4-face numbers:

${\begin{array}{l}\displaystyle {E_{\{P_{4}^{(3)}(n)\}}(x)=\;?.}\end{array}}$ For the 8-face and 6-face numbers:

${\begin{array}{l}\displaystyle {E_{\{P_{6}^{(3)}(n)\}}(x)=\;?;}\end{array}}$ ${\begin{array}{l}\displaystyle {E_{\{P_{8}^{(3)}(n)\}}(x)=\;?.}\end{array}}$ For the 20-face and 12-face numbers:

${\begin{array}{l}\displaystyle {E_{\{P_{12}^{(3)}(n)\}}(x)=\;?;}\end{array}}$ ${\begin{array}{l}\displaystyle {E_{\{P_{20}^{(3)}(n)\}}(x)=\;?.}\end{array}}$ Dirichlet generating function

${\begin{array}{l}\displaystyle {D_{\{P_{N_{0}}^{(3)}(n)\}}(x)=\;?.}\end{array}}$ For the 4-face numbers:

${\begin{array}{l}\displaystyle {D_{\{P_{4}^{(3)}(n)\}}(x)=\;?.}\end{array}}$ For the 8-face and 6-face numbers:

${\begin{array}{l}\displaystyle {D_{\{P_{6}^{(3)}(n)\}}(x)=\;?;}\end{array}}$ ${\begin{array}{l}\displaystyle {D_{\{P_{8}^{(3)}(n)\}}(x)=\;?.}\end{array}}$ For the 20-face and 12-face numbers:

${\begin{array}{l}\displaystyle {D_{\{P_{12}^{(3)}(n)\}}(x)=\;?;}\end{array}}$ ${\begin{array}{l}\displaystyle {D_{\{P_{20}^{(3)}(n)\}}(x)=\;?.}\end{array}}$ Order of basis

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and
 k
 k
-gon numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found. Joseph Louis Lagrange proved the square case (known as the four squares theorem) in 1770 and Gauss proved the triangular case in 1796. In 1813, Cauchy finally proved the horizontal generalization that every nonnegative integer can be written as a sum of
 k
 k
-gon numbers (known as the polygonal number theorem), while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem). A nonempty subset
 A
of nonnegative integers is called a basis of order
 g
if
 g
is the minimum number with the property that every nonnegative integer can be written as a sum of
 g
elements in
 A
. Lagrange’s sum of four squares can be restated as the set
 {n 2 | n = 0, 1, 2, …}
of nonnegative squares forms a basis of order 4.
Theorem. (Cauchy)

For every
 k   ≥   3
, the set
 {P (k, n) | n = 0, 1, 2, …}
of
 k
-gon numbers forms a basis of order
 k
, i.e. every nonnegative integer can be written as a sum of
 k
 k
-gon numbers.

Proof. PROOF GOES HERE. (Provide proof: PROOF GOES HERE. □)
We note that polygonal numbers are two dimensional analogues of squares. Obviously, cubes, fourth powers, fifth powers, ... are higher dimensional analogues of squares. In 1770, Waring stated without proof that every nonnegative integer can be written as a sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 1909, Hilbert proved that there is a finite number
 g (d)
such that every nonnegative integer is a sum of
 g (d)
 d
th powers, i.e. the set
 {n d | n = 0, 1, 2, …}
of
 d
th powers forms a basis of order
 g (d)
. The Hilbert-Waring problem is concerned with the study of
 g (d)
for
 d   ≥   2
. This problem was one of the most important research topics in additive number theory in last 90 years, and it is still a very active area of research.

Differences

$P_{N_{0}}^{(3)}(n)-P_{N_{0}}^{(3)}(n-1)=?$ Partial sums

The partial sums correspond to 4-dimensional Platonic hyperpyramidal numbers.

$\sum _{n=0}^{m}P_{N_{0}}^{(3)}(n)=?\,$ Partial sums of reciprocals

$\sum _{n=0}^{m}{\frac {1}{P_{N_{0}}^{(3)}(n)}}=?$ Sum of reciprocals

${\begin{array}{l}\displaystyle {\sum _{n=0}^{\infty }{\frac {1}{P_{N_{0}}^{(3)}(n)}}=\;?}\end{array}}$ The sum of reciprocals
 ∞

 n  = 1
 1 P  (3) 6(n)
= ζ (3)
can be interpreted as
 1 p
, where
p =
 1 ζ (3)
is the probability that three random integers
 x
,
 y
and
 z
are coprime.

Table of formulae and values

 N0, N1
and
 N2
are the number of vertices (0-dimensional), edges (1-dimensional) and faces (2-dimensional) respectively, where the faces are the actual facets. The regular Platonic numbers are listed by increasing number
 N0
of vertices.
Platonic numbers formulae and values
Rank

 r

 N0(2 r − 0 r ) + 4
Name

 (N0, N1, N2)

Schläfli
symbol
Formulae

 P  (3) N0(n)
Generating
function

 G{P  (3) N0(n)}(x)
0 1 2 3 4 5 6 7 8 9 10 11 12 A-number
0 4 Tetrahedral

(4, 6, 4)

{3, 3}

${\binom {n+2}{3}}$ ${\frac {n^{(3)}}{3!}}$ ${n(n+1)(n+2)} \over {6}\,$ ${x(1+0x+0x^{2})} \over {(1-x)^{4}}\,$ ${x} \over {(1-x)^{4}}\,$ 0 1 4 10 20 35 56 84 120 165 220 286 364 A000292
1 6 Octahedral

(6, 12, 8)

{3, 4}

${\binom {n+2}{3}}+2{\binom {n+1}{3}}+{\binom {n}{3}}\,$ $P_{4}^{(3)}(n)+2\ P_{4}^{(3)}(n-1)+P_{4}^{(3)}(n-2)\,$ $Y_{5}^{(3)}(n)+Y_{5}^{(3)}(n-1)\,$ ${n(2n^{2}+1)} \over {3}\,$ ${x(1+2x+x^{2})} \over {(1-x)^{4}}\,$ ${x(1+x)^{2}} \over {(1-x)^{4}}\,$ 0 1 6 19 44 85 146 231 344 489 670 891 1156 A005900
2 8 Cube

(8, 12, 6)

{4, 3}

${\binom {n+2}{3}}+4{\binom {n+1}{3}}+{\binom {n}{3}}\,$ $P_{4}^{(3)}(n)+4\ P_{4}^{(3)}(n-1)+P_{4}^{(3)}(n-2)\,$ $n^{3}\,$ ${x(1+4x+x^{2})} \over {(1-x)^{4}}\,$ 0 1 8 27 64 125 216 343 512 729 1000 1331 1728 A000578
3 12 Icosahedral

(12, 30, 20)

{3, 5}

${\binom {n+2}{3}}+8{\binom {n+1}{3}}+6{\binom {n}{3}}\,$ $P_{4}^{(3)}(n)+8\ P_{4}^{(3)}(n-1)+6\ P_{4}^{(3)}(n-2)\,$ $n(5P_{3}^{(2)}(n-1)+1)\,$ ${n(5n^{2}-5n+2)} \over {2}\,$ ${x(1+8x+6x^{2})} \over {(1-x)^{4}}\,$ 0 1 12 48 124 255 456 742 1128 1629 2260 3036 3972 A006564
4 20 Dodecahedral

(20, 30, 12)

{5, 3}

${\binom {n+2}{3}}+16{\binom {n+1}{3}}+10{\binom {n}{3}}\,$ $P_{4}^{(3)}(n)+16\ P_{4}^{(3)}(n-1)+10\ P_{4}^{(3)}(n-2)\,$ $n(9P_{3}^{(2)}(n-1)+1)\,$ ${n(9n^{2}-9n+2)} \over {2}\,$ ${n(3n-1)(3n-2)} \over {2}\,$ ${\binom {3n}{3}}$ $P_{4}^{(3)}(3n-2)\,$ $n\ \,_{c}P_{9}^{(2)}(n)\,$ $n\ P_{3}^{(2)}(3n-2)\,$ ${x(1+16x+10x^{2})} \over {(1-x)^{4}}\,$ 0 1 20 84 220 455 816 1330 2024 2925 4060 5456 7140 A006566

Table of related formulae and values

 N0, N1
and
 N2
are the number of vertices (0-dimensional), edges (1-dimensional) and faces (2-dimensional) respectively, where the faces are the actual facets. The regular Platonic numbers are listed by increasing number
 N0
of vertices.
Platonic numbers related formulae and values
Rank

 r

 N0

 (2 r  −  0 r  ) + 4
Name

 (N0, N1, N2)

Schläfli symbol
Order
of basis

 g{P  (3) N0(n)}

 N0 + ?
Differences

 P  (3) N0(n)  −  P  (3) N0(n  −  1)
Partial sums

 m ∑ n  = 1
P  (3)N0(n)
Partial sums
of reciprocals

 m ∑ n  = 1

 1 P  (3) N0(n)
Sum of Reciprocals

 ∞ ∑ n  = 1

 1 P  (3) N0(n)
0 4 Tetrahedral

(4, 6, 4)
{3, 3}
$5?\,$ $(N_{0}+1?)\,$ ${\frac {n(n+1)}{2}}\,$ $\,$ $\,$ ${\frac {3}{2}}$ 1 6 Octahedral

(6, 12, 8)
{3, 4}
$7?\,$ $(N_{0}+1?)\,$ $2n^{2}-2n+1\,$ ${\frac {4n(n-1)+2}{2}}\,$ $\,$ $\,$ $3{\bigg [}\gamma +\Re \psi {\bigg (}{\frac {i}{\surd {2}}}{\bigg )}{\bigg ]}$ Base 10: A175577
2 8 Cube

(8, 12, 6)
{4, 3}
$9\,$ $(N_{0}+1)\,$ $3n^{2}-3n+1\,$ ${\frac {6n(n-1)+2}{2}}\,$ $\,$ $\,$ $\zeta (3)\,$ Base 10: A002117
3 12 Icosahedral

(12, 30, 20)
{3, 5}
$15?\,$ $(N_{0}+3?)\,$ ${\frac {15n^{2}-25n+12}{2}}\,$ ${\frac {5n(3n-5)+12}{2}}\,$ $\,$ $\,$ $\gamma +\Re \psi {\big (}{\frac {1}{2}}+{\frac {i{\sqrt {15}}}{10}}{\big )}+{\frac {{\sqrt {15}}\pi }{6}}\tanh {\big (}{\frac {{\sqrt {15}}\pi }{10}}{\big )}$ Base 10: A175578
4 20 Dodecahedral

(20, 30, 12)
{5, 3}
$22?\,$ $(N_{0}+2?)\,$ ${\frac {27n^{2}-45n+20}{2}}\,$ ${\frac {9n(3n-5)+20}{2}}\,$ $\,$ $\,$ ${\frac {\pi {\sqrt {3}}-3\log {3}}{2}}$ $\approx 1.0727806\ldots \,$ Table of sequences

Platonic numbers sequences
 N0
 P  (3) N0(n), n   ≥   0.
A-number
4
{0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, ...}
A000292
6
{0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, ...}
A005900
8
{0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, ...}
A000578
12
{0, 1, 12, 48, 124, 255, 456, 742, 1128, 1629, 2260, 3036, 3972, 5083, 6384, 7890, 9616, 11577, 13788, 16264, 19020, 22071, 25432, 29118, 33144, 37525, 42276, 47412, ...}
A006564
20
{0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, 14190, 17296, 20825, 24804, 29260, 34220, 39711, 45760, 52394, 59640, 67525, 76076, ...}
A006566